In other words, any invariant measure is supported in \(\tilde{V} = V \times H\) .

The idea of the proof was inspired by the work [1]}, in which the authors derived a similar result for the invariant measure emerging in stochastic primitive equations. As it was shown in [2]} (see proof of Theorem 5.1), if \(v = (u,w)\)  is a solution of (REF ), then
\(\int _{0}^{T} \Vert v\Vert _{\tilde{V}}^2 dt = \int _{0}^{T} (\Vert u\Vert _{V}^2 + \Vert w\Vert ^2_{H}) dt < K_1 \Vert v_0\Vert _{\tilde{H}}^2 + K_2 T\) 
